Average Error: 58.1 → 0.7
Time: 9.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r158706 = 0.5;
        double r158707 = re;
        double r158708 = cos(r158707);
        double r158709 = r158706 * r158708;
        double r158710 = 0.0;
        double r158711 = im;
        double r158712 = r158710 - r158711;
        double r158713 = exp(r158712);
        double r158714 = exp(r158711);
        double r158715 = r158713 - r158714;
        double r158716 = r158709 * r158715;
        return r158716;
}
↓
double f(double re, double im) {
        double r158717 = 0.5;
        double r158718 = re;
        double r158719 = cos(r158718);
        double r158720 = r158717 * r158719;
        double r158721 = 0.3333333333333333;
        double r158722 = im;
        double r158723 = 3.0;
        double r158724 = pow(r158722, r158723);
        double r158725 = r158721 * r158724;
        double r158726 = 0.016666666666666666;
        double r158727 = 5.0;
        double r158728 = pow(r158722, r158727);
        double r158729 = r158726 * r158728;
        double r158730 = 2.0;
        double r158731 = r158730 * r158722;
        double r158732 = r158729 + r158731;
        double r158733 = r158725 + r158732;
        double r158734 = -r158733;
        double r158735 = r158720 * r158734;
        return r158735;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Final simplification0.7

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))